14) Polar Coordinates
Butterflies
Polar equations of the form r =1+sin(nTheta)+(cos2ntheta)^2
> plot([1+sin(2*theta)+(cos(4*theta))^2,theta,theta =0..2*Pi], coords=polar);
> plot([1+sin(3*theta)+(cos(6*theta))^2,theta,theta =0..2*Pi], coords=polar);
Arc Length in Polar Coordinates
If r = f(theta), the arc length from theta =alpha to theta =beta is given by
> restart;
> LP:=Int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=alpha..beta);
Find the arc length of r=1+cos (theta) without using symmetry.
> f:=(theta)->1+cos(theta);
> plot([f(theta),theta,theta=0..2*Pi], coords=polar);
> Int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=0..2*Pi);
> simplify(%);
> value(%);
Do the same problem using symmetry
> 2*int (sqrt(f(theta)^2+D(f)(theta)^2) ,theta=0..Pi);